Let \(\varrho _{0}=P_{0}\) . As in the proof of Theorem REF , we have the following short exact sequence
\(0\rightarrow \mathbb {F}_{\varrho }\rightarrow \Delta ^*(\mathrm {pr}_{-}^*\mathbb {F}_{-}\oplus \mathrm {pr}_{+}^*\mathbb {F}_{+})\rightarrow \iota _{\mathcal {D}*}\iota _{\mathcal {D}}^*\mathbb {F}_{\varrho }\rightarrow 0.\) 
Given \(\gamma \in A_{\mathrm {orb}}^*(\mathcal {X})\) , the operator \(\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma )\)  can be similarly defined as in Section 6.2. Notice that the refined Gysin homomorphism defined in the case of Deligne-Mumford stacks (see [[1]}, [2]}]) has the formal properties of commuting with proper pushforward and flat pull back, and commuting with intersection products with Chern classes as in [[3]}, Proposition 6.3].
If \(\gamma _{\pm }\)  is disjoint with \(\mathcal {D}\) , by Theorem REF , we have
\(&&\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma )\cdot i_{0}^![\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}\\&=&\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\pi _{2*}\left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\mathbb {F}_{\varrho })\cdot \iota ^*\bar{\pi }_{1}^*\gamma \cap \pi _{2}^*\varsigma _{\varrho *}\Delta ^!([\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\times [\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir})\right)\\&=&\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\varsigma _{\varrho *}\pi _{2*}\left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\Delta ^*\mathrm {pr}_{-}^*\mathbb {F}_{-})\cdot \iota ^*\bar{\pi }_{1}^*\gamma _{-}\cap \pi _{2}^*\Delta ^!([\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\times [\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir})\right)\\&&+\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\varsigma _{\varrho *}\pi _{2*}\left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\Delta ^*\mathrm {pr}_{+}^*\mathbb {F}_{+})\cdot \iota ^*\bar{\pi }_{1}^*\gamma _{+}\cap \pi _{2}^*\Delta ^!([\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\times [\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir})\right)\\&=&\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\varsigma _{\varrho *}\pi _{2*}\Delta ^!\left(\left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\mathbb {F}_{-})\cdot \iota _{-}^*\bar{\pi }_{1,-}^*\gamma _{-}\cap \pi _{2,-}^*[\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\right)\times \pi _{2,+}^*[\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir}\right)\\&&+\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\varsigma _{\varrho *}\pi _{2*}\Delta ^!\left(\pi _{2,-}^*[\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\times \left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\mathbb {F}_{+})\cdot \iota _{+}^*\bar{\pi }_{1,+}^*\gamma _{+}\cap \pi _{2,+}^*[\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir}\right)\right)\\&=&\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\varsigma _{\varrho *}\Delta ^!\left(\left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\gamma _{-})\cdot [\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\right)\times [\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir}\right)\\&&+\sum _{\varrho \in \Lambda _{P}^{\mathrm {spl}}}\varsigma _{\varrho *} \Delta ^!\left([\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\times \left(\widetilde{\mathrm {ch}}^\mathrm {orb}_{k+2}(\gamma _{+})\cdot [\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir}\right)\right)\\\) 
where the map \(\pi _{i}\)  is the projections from \(I_{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {X}_{0}^{\dagger ,\varrho }\times _{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \)  to the \(i\) -th factor for \(i=1,2\) , the map \(\bar{\pi }_{1}\)  is the composition map of \(\pi _{1}\)  with the natural map \(I_{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {X}_{0}^{\dagger ,\varrho }\rightarrow I\mathcal {X}_{0}\)  and \(\iota : I_{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {X}_{0}^{\dagger ,\varrho }\rightarrow I_{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {X}_{0}^{\dagger ,\varrho }\)  is the canonical involution. Similarly, the map \(\pi _{i,\pm }\)  is the projections from \(I_{\mathfrak {A}^{\varrho _{\pm },\varrho _{0}}}\mathfrak {Y}^{\varrho _{\pm },\varrho _{0}}\times _{\mathfrak {A}^{\varrho _{\pm },\varrho _{0}}}\mathfrak {PT}^{\varrho _{\pm },\varrho _{0}}_{\mathfrak {Y}_{\pm }/\mathfrak {A}}\)  to the \(i\) -th factor (\(i=1,2\) ), the map \(\bar{\pi }_{1,\pm }\)  is the composition map with \(I_{\mathfrak {A}^{\varrho _{\pm },\varrho _{0}}}\mathfrak {Y}^{\varrho _{\pm },\varrho _{0}}\rightarrow I\mathcal {Y}_{\pm }\) , and \(\iota _{\pm }: I_{\mathfrak {A}^{\varrho _{\pm },\varrho _{0}}}\mathfrak {Y}^{\varrho _{\pm },\varrho _{0}}\rightarrow I_{\mathfrak {A}^{\varrho _{\pm },\varrho _{0}}}\mathfrak {Y}^{\varrho _{\pm },\varrho _{0}}\)  is the canonical involution.
On the other hand, we have \(\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma )\cdot i_{c}^![\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}=\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma )\cdot [ \mathrm {PT}_{\mathcal {X}_{c}/\mathbb {C}}^P]^\mathrm {vir}\)  by Theorem REF .
Applying \(\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma _{i})\)  successively to both \(i_{0}^![\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}\)  and \(i_{c}^![\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}\) , and as in the proof of [[4]}, Theorem 6.8], using the following identity (where \([\cdot ]_{1}\in A_{1}(\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}})\)  means taking degree one part)
\(\deg i_{c}^!\left[\prod _{i=1}^r\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma _{i})\cdot [\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}\right]_{1}=\deg i_{0}^!\left[\prod _{i=1}^r\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma _{i})\cdot [\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}]^\mathrm {vir}\right]_{1}\) 
and the Kunneth decomposition of \([\Delta ]\) , the proof is completed by using Definitions REF  and REF .
